118 



Sii* F. Pollock on the ^'Mysteries of Numbers^' [ Apr. 19, 



tive roots increased upwards and decreased downwards, the result will be 

 as in the Table below : — 





Order of 





Algebraic 



Sums 



Order 





of 



Eoots. 



sums of 



of squares 



of 





terms. 





roots. 



of roots. 



terms. 







4 5 7 











12 



-13-9-4 3 

 4 5 7 



-23 



275 



12 





10 



8 



-12-8-3 4 



4 5 7 

 -11-7-2 5 



4 5 7 



-19 

 -15 



233 

 199 



10 



8 





6 



— 10—6 — 1 6 



4 5 7 



— 11 



173 



6 





4 



— 9—5 7 

 4 5 7 



— 7 



155 



4 





2 



- 8-4 1 8 



- 3 



145 



2 







4 5 7 









Centre 



1 



— 7—3 2 9 



1 



143 



1 







4 5 7 











3 



- 6-2 3 10 



4 5 7 



5 



149 



3 





5 



- 5 — 1 4 11 



4 5 7 



9 



163 



5 





/ 



- 4 5 12 



4 5 7 



13 



185 



7 





9 



— 3 1 6 13 



4 5 7 



17 



215 - 



9 





11 



- 2 2 7 14 



4 5 7 



21 



253 



11 





13 



- 1-3 8 15 

 &c. &c. 



25 

 &c. 



299 



13 



It will be observed that the terms of the series 143, 145, 149, 155, 

 &c. increase by 2, 4, 6, 8, . . (2 n). In the column of the sums of the 

 roots 1, 5, 9, 13, &c. increase by 4. 



1,-3, — 7, — 11 decrease by 4; the differences of the roots, if arranged 

 in the order of their numerical value, is always the same throughout the 

 series. 



Note. — In the remainder of this paper every odd number that becomes 

 a term in any of the series is expressed by the roots the sum of whose 



squares form the number itself; 



2, 3, 6, 8 



means that the number 



occupying that division or small square is 113 = (2^ + 3'- + 6^8'); a figure 

 (or collection of figures) representing merely the arithmetical value is put 



into a circle, thus: 



13 



2, 3, 6, 8 



